It would appear that he was detailed on extra duty to serve as a musician in the hospital band. I'm not aware that bands were assigned to hospitals, but it would make sense.

I doubt rather much one would find an individual list of 37th Infantry men. The War Department copyist who recorded this information on William took it from some large list of soldiers, how they were categorized is not known. However, this tabulation was made some 100 years ago. The original list would be buried somewhere within the bowels of the National Archives.